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In this paper we present a systematic and rigorous method for calculating thediffusion tensor for a Brownian particle moving in a periodic potential whichis valid in arbitrary dimensions and for all values of the dissipation. We usethis method to obtain an explicit formula for the diffusion coefficient in onedimension which is valid in the underdamped limit, and we also obtain higherorder corrections to the Lifson-Jackson formula for the diffusion coefficientin the overdamped limit. A numerical method for calculating the diffusioncoefficient is also developed and is shown to perform extremely well for allvalues of the dissipation.

DIFFUSIVE TRANSPORT IN PERIODIC POTENTIALS: UNDERDAMPED DYNAMICS